Lesson 6

Construction Techniques 4: Parallel and Perpendicular Lines

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Problem 1

Which of the following constructions would help to construct a line passing through point \(C\) that is perpendicular to the line \(AB\)?

Line segment. Point A is on the left side, point B on the right. Point C is above the line segment, closer to Point A.
A:

Construction of an equilateral triangle with one side \(AB\)

B:

Construction of a hexagon with one side \(BC\)

C:

Construction of a perpendicular bisector through \(C\)

D:

Construction of a square with one side \(AB\)

Solution

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Problem 2

Two distinct lines, \(\ell\) and \(m\), are each perpendicular to the same line \(n\). Select all the true statements.

A:

Lines \(\ell\) and \(m\) are perpendicular.

B:

Lines \(\ell\) and \(n\) are perpendicular.

C:

Lines \(m\) and \(n\) are perpendicular.

D:

Lines \(\ell\) and \(m\) are parallel.

E:

Lines \(\ell\) and \(n\) are parallel.

F:

Lines \(m\) and \(n\) are parallel.

Solution

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Problem 3

This diagram is a straightedge and compass construction of the bisector of angle \(BAC\). Only angle \(BAC\) is given. Explain the steps of the construction in order. Include a step for each new circle, line, and point.

Three circles

Solution

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(From Unit 1, Lesson 5.)

Problem 4

This diagram is a straightedge and compass construction of a line perpendicular to line \(AB\) passing through point \(C\). Which segment has the same length as segment \(EA\)?

Three circles.
A:

\(EC\)

B:

\(ED\)

C:

\(BE\)

D:

\(BD\)

Solution

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(From Unit 1, Lesson 5.)

Problem 5

This diagram is a straightedge and compass construction. Which triangle is equilateral? Explain how you know.

Seven congruent overlapping circles.

Solution

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(From Unit 1, Lesson 4.)

Problem 6

In the construction, \(A\) is the center of one circle, and \(B\) is the center of the other. Name the segments in the diagram that have the same length as segment \(AB\).

Two circles, centered at A and B, each pass through the center of the other and intersect at C and D. Line AB extends horizontally across both circles. Radii AC, BC, AD and BD form rhombus ACBD.

 

Solution

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(From Unit 1, Lesson 2.)

Problem 7

This diagram is a straightedge and compass construction. \(A\) is the center of one circle, and \(B\) is the center of the other. 

  1. Name a pair of perpendicular line segments.
  2. Name a pair of line segments with the same length.
two intersecting circles

Solution

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(From Unit 1, Lesson 3.)

Problem 8

\(A\), \(B\), and \(C\) are the centers of the 3 circles. Select all the segments that are congruent to \(AB\).

Three intersecting circles.
A:

\(HF\)

B:

\(HA\)

C:

\(CE\)

D:

\(CD\)

E:

\(BD\)

F:

\(BF\)

Solution

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(From Unit 1, Lesson 4.)